Let $A=\{1,2,3\}$. The number of relations on $A$ that are reflexive and symmetric is:

Solution

1. For a set $A$ with $n=3$ elements, a reflexive relation must contain all diagonal elements $(1,1),(2,2),(3,3)$.
2. A symmetric relation is determined by the remaining $n(n-1)/2$ pairs, which for $n=3$ equals $3(2)/2 = 3$ pairs: $(1,2),(1,3),$ and $(2,3)$.
3. Each of these 3 pairs can either be included or excluded independently, resulting in $2^3=8$ possible relations.
4. Hence the answer is (A).